Automorphic orbits in free groups: recent progress

Mathematics > Group Theory [Submitted on 1 Oct 2025 (v1), last revised 17 Jun 2026 (this version, v4)] Title:Automorphic orbits in free groups: recent progress View PDF HTML (experimental)Abstract:In this survey, we describe recent progress on asymptotic properties of various automorphic orbits in free groups. In particular, we address the problem of counting potentially positive elements of a given length. We also discuss complexity (worst-case, average-case, and generic-case) of Whitehead's automorphism problem and relevant properties of automorphic orbits, including orbit-blocking words. Submission history From: Vladimir Shpilrain [view email] [via Murray Elder as proxy][v1] Wed, 1 Oct 2025 13:35:08 UTC (10 KB) [v2] Thu, 2 Oct 2025 15:36:30 UTC (10 KB) [v3] Wed, 8 Oct 2025 15:39:21 UTC (10 KB) [v4] Wed, 17 Jun 2026 19:54:04 UTC (18 KB) References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.